Question

Solve the inequality. Express the solution set in​ set-builder notation and interval notation. Graph the solution set on a real number line.
5(x−9)>4(x−9)+x

Answer 1

To solve the inequality \(5(x - 9) > 4(x - 9) + x\), we will first expand and simplify both sides.

1. Expand both sides:
- Left side:
\[
5(x - 9) = 5x - 45
\]
- Right side:
\[
4(x - 9) + x = 4x - 36 + x = 5x - 36
\]

2. Now, the inequality becomes:
\[
5x - 45 > 5x - 36
\]

3. Next, we will subtract \(5x\) from both sides:
\[
-45 > -36
\]

This inequality does not depend on \(x\) and simplifies to a false statement:

\(-45\) is not greater than \(-36\), meaning there are no values of \(x\) that satisfy the inequality. Therefore, the solution set is empty.

**Expressing the Solution:**

- In **set-builder notation**, the solution set is:
\[
\{ x \mid \text{no solution} \}
\]

- In **interval notation**, the solution is represented as:
\[
\emptyset
\]

**Graphing the Solution Set:**
Since there are no solutions, there would be no shading on the number line.

You can simply indicate that the entire line has no region shaded, or you can represent it as a closed interval showing no values:
- (no shading on the line)

In conclusion, the solution to the inequality is the empty set both in set-builder and interval notation, and it is represented on the number line with no shaded regions.